时间序列-1-随机变量，收敛定理

2020-03-16 本文已影响0人蒋子义

Probability Space

$(\Omega,\mathcal{F}, \mathbb{P})$
Here $\Omega$ is a set and $\mathcal{F}$ is a $\sigma$ -algebra (of subsets of $X$ ) which satisfies:

$\phi \in \mathcal{F}$
$E \in \mathcal{F} \Rightarrow E^x \in \mathcal{F}$
$E_1, E_2, .., \in \mathcal{F}\Rightarrow\bigcup_{n=1}^{\infty}E_n \in\mathcal{F}$

$\mathbb{P}$ is a function satisfies:

$\mathbb{P}: \mathcal{F} \rightarrow [0,1]$
$\mathbb{P}(\phi) = 0$
$\mathbb{P}$ is countable additive

Remark:

the sequence of $(\Omega,\mathcal{F}, \mathbb{P})$ is : $\Omega \Rightarrow \mathcal{F}\Rightarrow\mathbb{P}$
the difference between Probability measure and general measure is $\forall A \in \mathcal{F}, \mathbb{P}(A)$ is bounded

Random variable and measurable function

A function $X$ : $\Omega\rightarrow\mathbb{R}$ is called a random variable if for every $x\in\mathbb{R}$ , the preimage $\{ X\le x \}$ = $X^{-1}((-\infty,x])$ is an event (belongs to the sigma-field $\mathcal{F}$ ).

Remark:

$\mathbb{P}(|X|=\infty)=0$ . If $\mathbb{P}(|X|=\infty)>0$ , it is a generalized random variable(or random map).

Lebesgue's montone convergence theorem

If $X_n$ is a sequence of nonnegative random variadble such that $X_n \le X_{n+1}$ and $X_n \rightarrow X$ a.s. Then
$EX_n \rightarrow EX$

Remark

Since $X_n$ is montonic, it is guaranteed to exist a generalized random variable $X$ that $X_n \rightarrow X$ a.s. It works for the situation that $X$ is a generalized random variable
The montone of $X_n$ guarantees that $EX$ exists(maybe $\infty$ )
Proof:
1. $EX_n\le EX$ due to $X_n \le X$ a.s.
2. $EX_n\ge EX - \epsilon$ by using a simple random variable $Z$ controled by $X$ . Prove $\lim_n EX_n \le Z\alpha$ , $\alpha$ is any number less than 1.

Fatou's lemma

If $X_1, X_2,...,X_n$ are nonnegative random variables, then
$E\liminf_{n\rightarrow\infty}X_n \le \liminf_{n\rightarrow\infty}EX_n$

proof: let $Y_n=\inf_{k \ge n}X_k$ . $EY_n \rightarrow E\liminf_{n\rightarrow\infty}X_n \le \liminf_{n\rightarrow\infty}EX_n$

Lebesgue's dominated convergence theorem

IF $\{X_n\}$ is a sequence of random variable, $X_n \rightarrow X$ a.s. and there exists an intergrable random $Y$ such that $|X_n| \le Y$ , then
$E|X_n-X|\rightarrow 0$

proof: Using Fatou's lemma for $Z_n = 2Y-|X_n-X|$

Different type of convergence

convergence in distribution

convergence in probability

convergence a.s.

$L^p$ convergence

Remark

$\begin{aligned} (4) &\Rightarrow (2)\\ (3) &\Rightarrow (2)\\ (2) &\Rightarrow (1)\\ L^p &\Rightarrow L^q (p>q) \end{aligned}$

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